In physics, the main objective in spin glasses is to understand
the strange magnetic properties of alloys.
Yet the models invented to explain the observed phenomena are of a rather
fundamental nature in mathematics. In this talk, we will focus on one of the most important mean field
models,
called the Sherrington-Kirkpatrick model,
and discuss its disorder chaos problem. Using Guerra's replica
symmetric-breaking bound, we present a mathematically rigorous proof for this problem.

The talk will review some elementary, but amusing, results concerning
surface spectra for discrete Laplacians on half-planes with a boundary.In particular, interesting differences arise for square lattices with
straight boundaries between the case where the boundary has the same
direction of the lattice and the one where the boundary is slanted
at an angle of 45 degrees to the direction of the lattice. This
is joint work with Y. Kreimer.

Since the discovery of quasicrystals by Schechtman et. al.
in 1984, quasi-periodic models in mathematical physics have formed an
active area of research. In particular, effects of quasi-periodicity
were investigated in a widely studied model of magnetism: the Ising
model (quantum and classical). Numerical and some analytic results
began to appear in the late '80s; however, most interesting
(numerical) results hitherto remained rigorously unconfirmed.Most of
the previous results relied on a connection with hyperbolic dynamical
systems.It is our aim to rigorously confirm previous numerical
observations, as well as to prove new results, by exploiting further
the aforementioned connection. In particular, we'll prove
multi-fractal structure of the energy spectrum of one-dimensional
quantum quasi-periodic Ising models. We'll also discuss its fractal
dimensions and measure.